We report heat transfer and temperature profile measurements in laboratory experiments of rapidly rotating convection in water under intense thermal forcing (Rayleigh number Ra as high as ∼1013) and unprecedentedly strong rotational influence (Ekman numbers E as low as 10-8). Measurements of the midheight vertical temperature gradient connect quantitatively to predictions from numerical models of asymptotically rapidly rotating convection, separating various flow phenomenologies. Past the limit of validity of the asymptotically reduced models, we explore a regime we refer to as rotationally influenced turbulence, where rotation is important but not as dominant as in the known geostrophic turbulence regime. The temperature gradients collapse to a Rayleigh-number scaling as Ra-0.2 in this regime. It is bounded from above by a critical convective Rossby number Ro∗=0.06 independent of domain aspect ratio Γ, clearly distinguishing it from well-studied rotation-affected convection.